## From hyperelliptic to superelliptic curves

There are two different ways of studying an algebraic curves; function fields $L/k(x)$ or coverings of $\P^1$.
The goal of this book is the study of algebraic curves via coverings in the tradition of Riemann, Clebsch, Hurwitz,
Severi, Grothendieck, et al.

We aim to highlight the theories that can be extended and all the open problems that come with this generalization.
We investigate the correspondence from the group theory data of the cover $f: \X \to \P^1$, via Riemann Existence
Theorem (RET), to the field of moduli of $\X$, relations among the thetanulls on $\Jac (\X)$ and the branch points
of $f: \X \to \P^1$, the Hurwitz spaces of coverings with ramification structure as that of $f$. This enables us to use
the full machinery of group theory to study covers and get more information about the arithmetic aspects of curves.

## Contents

Motivation

#### Part 1: Preliminaries

1. Coverings
• Covering spaces and the fundamental group
• Deck transformations
• The punctured sphere and ramification type
• Riemann Existence Theorem
• Riemann surfaces and the Hurwitz formula
2. Function fields
• Function fields
• Divisors
• Extensions
• Integral closure and locality
• Hurwitz genus formula and the different
• Galois extensions
• Cyclic extensions
• Finite extensions of $\mathbb C(x)$
• Branch points and conjugacy classes
3. Curves
• Affine and projective spaces
• Forms
• Projective varieties
• Projective curves and function fields
• Intersection of curves
• Morphisms of curves and branched coverings
• The Hurwitz genus formula
• Extensions of function fields, morphisms, coverings
4. Automorphisms of curves
• Automorphism groups, $G$-actions, stabilizers
• Cyclic $n$-gonal curves
• Weierstrass points
• Automorphisms
• Hyperelliptic curves
• Superelliptic curves

#### Part 2: Moduli spaces

5. Hurwitz spaces
6. Moduli space of curves
7. Describing points in the moduli
8. Weighted moduli spaces
9. Equations of curves
10. #### Part 3: Superelliptic Jacobian varieties

11. Theta functions
12. Jacobian varieties
13. Complex multiplication
14. #### Part 4: Arithemtic

15. Obstruction in the moduli space
16. Reduction theory of binary forms
17. Theory of heights
18. Minimal models
19. Mordell-Weil group for superelliptic Jacobians